A0157
Title: Post-processed posteriors for high-dimensional covariances
Authors: Jaeyong Lee - Seoul National University (Korea, South) [presenting]
Abstract: Bayesian inference of high-dimensional covariance matrices with structural assumptions, such as banded and bendable covariances, is considered, and a post-processed posterior is proposed. The post-processing of the posterior consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior, which does not satisfy any structural restrictions. In the second step, the posterior samples are transformed to satisfy the structural restriction through a post-processing function. The conceptually straightforward procedure of the post-processed posterior makes its computation efficient and can render interval estimators of functionals of covariance matrices. It is shown that it has nearly optimal minimax rates for banded and bendable covariances among all possible pairs of priors and post-processing functions. Additionally, a theorem on the credible set of the post-processed posterior under the finite dimension assumption is provided. It is proved that the expected coverage probability of the (1-a)100\% highest posterior density region of the post-processed posterior is asymptotically 1-a with respect to any conventional posterior distribution. It implies that the highest posterior density region of the post-processed posterior is, on average, a credible set of conventional posterior. The advantages of the post-processed posterior are demonstrated by a simulation study and a real data analysis.
